### Speaker

### Description

Strong symmetries of Fractional Brownian Motion (FBM) make its Hurst index $H$ the unifying number governing both its short (fractal dimension $2-H$) and long time (mean squared displacement $\propto t^{2H}$) properties. This fact restricts some of its applications—crucially it makes it impossible to describe an increasing number of regime switching anomalous diffusion systems in which the anomalous diffusion exponent and the diffusivity change as functions of time. These are, e.g., motions of a tracer in the changing viscoelastic environment of cells during their cycle, solutions under pressure and/or concentration changes, the motion of lipid molecules in cooling bilayer membranes or passive and active intracellular movement after treatment with chemicals. In order to overcome the limitations of FBM, multifractional Brownian motion (MFBM) models were created, initially motivated by terrain modelling. However, most of the existing types of MFBM use Hurst index as purely local measure of the path roughness and cannot be used as models of physical diffusion switching from one type of complex environment to another.

We will show how to overcome this limitation and present models of a particle diffusing in a complex environment for which conditions change in time and after the transition new displacements are governed by a new diffusivity and a new Hurst exponent while also keeping the memory of its history before the transition. We can describe both smooth and stepwise transitions. We obtain the mean squared displacement and correlations of these models and study their behaviour. Finally, we present estimation methods and discuss how this modelling approach can help with analysing experimental data.